1. Use Piecewise Functions
2.5
Use Piecewise Functions
Example 1
Evaluate a piecewise function
Evaluate the function when x = 3.
Solution
Because ______, use _______ equation.
second
Substitute ___ for x.
Simplify.

2. Use Piecewise Functions
2.5
Use Piecewise Functions
Checkpoint. Evaluate the function when x = -4 and x = 2.

3. Use Piecewise Functions
2.5
Use Piecewise Functions
Example 2
Graph a piecewise function
Find the x-coordinates for which there are points of discontinuity.
Graph
Solution
To the _____ of x = -1, graph y = -2x + 1. Use an _____ dot at (-1, ___ ) because the equation y = -2x + 1 __________ apply when x = -1.
left
open
does not
From x = -1 to x = 1, inclusive, graph y = ¼ x. Use _____ dots at both ( -1, ___ ) and ( 1, ___ ) because the equation y = ¼ x applies to both x = -1 and x = 1.
solid
- ¼

4. Use Piecewise Functions
2.5
Use Piecewise Functions
Example 2
Graph a piecewise function
Find the x-coordinates for which there are points of discontinuity.
Graph
Solution
To the right of x = 1, graph y = 3. Use an _____ dot at (1, ___ ) because the equation y = 3 __________ apply when x = 1.
open
does not
Examine the graph. Because there are gaps in the graph at x = _____ and x = ___, these are the x-coordinates for which there are points of _____________.
discontinuity

5. Use Piecewise Functions
2.5
Use Piecewise Functions
Checkpoint. Complete the following exercise.
Graph the following function and find the x-coordinates for which there are points of discontinuity.
Discontinuity at x = 0
and x = 2

6. Use Piecewise Functions
2.5
Use Piecewise Functions
Example 3
Write a piecewise function
Write a piecewise function for the step function shown. Describe any intervals over which the function is constant.
For x between ___ and ___, including x = 1, the graph is the line segment given by y = 1.
For x between ___ and ___, including x = 2, the graph is the line segment given by y = 2.
So, a _____________ _________ for the graph is as follows:
piecewise
For x between ___ and ___, including x = 3, the graph is the line segment given by y = 3.
function
The intervals over which the function is ___________ are ____________,
____________, ______________.
constant

7. Use Piecewise Functions
2.5
Use Piecewise Functions
Example 4
Write and analyze a piecewise function
Write the function as a piecewise function. Find any extrema as well as the rate of change of the function to the left and to the right of the vertex.
Graph the function. Find and label the vertex, one point to the left of the vertex, and one point to the right of the vertex. The graph shows one minimum value of ____, located at the vertex, and no maximum.

8. Use Piecewise Functions
2.5
Use Piecewise Functions
Example 4
Write and analyze a piecewise function
Write the function as a piecewise function. Find any extrema as well as the rate of change of the function to the left and to the right of the vertex.
Find linear equations that represent each piece of the graph.
Left of vertex:

9. Use Piecewise Functions
2.5
Use Piecewise Functions
Example 4
Write and analyze a piecewise function
Write the function as a piecewise function. Find any extrema as well as the rate of change of the function to the left and to the right of the vertex.
Find linear equations that represent each piece of the graph.
Right of vertex:

10. Use Piecewise Functions
2.5
Use Piecewise Functions
Example 4
Write and analyze a piecewise function
Write the function as a piecewise function. Find any extrema as well as the rate of change of the function to the left and to the right of the vertex.
So the function may be written as
The extrema is a ____________ located at the vertex ( -1, -2 ). The rate of change of the function is ____ when x < -1 and ___ when x > -1.
minimum

11. Use Piecewise Functions
2.5
Use Piecewise Functions
Checkpoint. Complete the following exercises.
Write a piecewise function for the step function shown. Describe any intervals over which the function is constant.
Constant intervals:

12. Use Piecewise Functions
2.5
Use Piecewise Functions
Checkpoint. Complete the following exercises.
Write the function as a piecewise function. Find any extrema as well as the rate of change to the left and to the right of the vertex.
minimum:
rate of change: